What options to grow object?
You could use the Mask modifier.
First you need to paint some weights to vertices in Weight Paint mode.
Go to Weight Paint mode and select your object. Add a vertex group and select Gradient tool. Pull the tool across your object and it will paint the weights as can be seen in the image below.

Next go to Object Mode and give your object a Mask Modifier. Pick the Vertex Group you created earlier and adjust the Threshold. Threshold can be keyframed to make the object appear and disappear.

The more subdivisions your object has the smoother the appearing is.